On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations

Alessandra Jannelli; Maria Paola Speciale; Alessandra Jannelli; Maria Paola Speciale

doi:10.3934/math.2021529

AIMS Mathematics

2021, Volume 6, Issue 8: 9109-9125. doi: 10.3934/math.2021529

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On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations

Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d'Alcontres 31, Messina, Italy

Received: 18 May 2021 Accepted: 16 June 2021 Published: 18 June 2021
MSC : 35R11, 22E70, 70G65, 34A08, 65L05

In this paper, we investigate the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations obtained by applying a procedure that combines the Lie symmetry analysis with the numerical methods. By Lie symmetries, the model, governed by two fractional differential equations defined in terms of the Riemann-Liouville fractional derivatives, is reduced into nonlinear fractional ordinary differential equations that, by introducing the Caputo derivative, are numerically solved by the implicit trapezoidal method. The solutions of the original model are computed by the inverse transformations. Numerical examples are performed in order to show the efficiency and the reliability of the proposed approach applied for solving a wide class of fractional models.
- time-fractional Reaction-diffusion equations,
- Riemann-Liouville fractional derivatives,
- Lie symmetries,
- Caputo derivatives,
- implicit trapezoidal method
Citation: Alessandra Jannelli, Maria Paola Speciale. On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations[J]. AIMS Mathematics, 2021, 6(8): 9109-9125. doi: 10.3934/math.2021529

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Abstract

In this paper, we investigate the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations obtained by applying a procedure that combines the Lie symmetry analysis with the numerical methods. By Lie symmetries, the model, governed by two fractional differential equations defined in terms of the Riemann-Liouville fractional derivatives, is reduced into nonlinear fractional ordinary differential equations that, by introducing the Caputo derivative, are numerically solved by the implicit trapezoidal method. The solutions of the original model are computed by the inverse transformations. Numerical examples are performed in order to show the efficiency and the reliability of the proposed approach applied for solving a wide class of fractional models.

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